Tauberian theorems for Cesàro summable double sequences

Ferenc Móricz

Studia Mathematica (1994)

  • Volume: 110, Issue: 1, page 83-96
  • ISSN: 0039-3223

Abstract

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( s j k : j , k = 0 , 1 , . . . ) be a double sequence of real numbers which is summable (C,1,1) to a finite limit. We give necessary and sufficient conditions under which ( s j k ) converges in Pringsheim’s sense. These conditions are satisfied if ( s j k ) is slowly decreasing in certain senses defined in this paper. Among other things we deduce the following Tauberian theorem of Landau and Hardy type: If ( s j k ) is summable (C,1,1) to a finite limit and there exist constants n 1 > 0 and H such that j k ( s j k - s j - 1 , k - s j - 1 , k + s j - 1 , k - 1 ) ≥ - H , j ( s j k - s j - 1 , k ) ≥ - H and k ( s j k - s j , k - 1 ) ≥ - H whenever j , k > n 1 , then ( s j k ) converges. We always mean convergence in Pringsheim’s sense. Our method is suitable to obtain analogous Tauberian results for double sequences of complex numbers or for those in an ordered linear space over the real numbers.

How to cite

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Móricz, Ferenc. "Tauberian theorems for Cesàro summable double sequences." Studia Mathematica 110.1 (1994): 83-96. <http://eudml.org/doc/216100>.

@article{Móricz1994,
abstract = {$(s_\{jk\}: j,k = 0,1,...)$ be a double sequence of real numbers which is summable (C,1,1) to a finite limit. We give necessary and sufficient conditions under which $(s_\{jk\})$ converges in Pringsheim’s sense. These conditions are satisfied if $(s_\{jk\})$ is slowly decreasing in certain senses defined in this paper. Among other things we deduce the following Tauberian theorem of Landau and Hardy type: If $(s_\{jk\})$ is summable (C,1,1) to a finite limit and there exist constants $n_1 > 0$ and H such that $jk(s_\{jk\} - s_\{j-1,k\} - s_\{j-1,k\} + s_\{j-1,k-1\}) ≥ -H$, $j(s_\{jk\} - s_\{j-1, k\}) ≥ -H$ and $k(s_\{jk\} - s_\{j,k-1\}) ≥ -H$ whenever $j,k > n_1$, then $(s_\{jk\})$ converges. We always mean convergence in Pringsheim’s sense. Our method is suitable to obtain analogous Tauberian results for double sequences of complex numbers or for those in an ordered linear space over the real numbers.},
author = {Móricz, Ferenc},
journal = {Studia Mathematica},
keywords = {double sequence; convergence in Pringsheim's sense; summability (C,1,1); (C,1,0) and (C,0,1); one-sided Tauberian condition of Landau and Hardy type; slow decrease; ordered linear space; Cesàro summability; Tauberian theorem; complex-valued sequences},
language = {eng},
number = {1},
pages = {83-96},
title = {Tauberian theorems for Cesàro summable double sequences},
url = {http://eudml.org/doc/216100},
volume = {110},
year = {1994},
}

TY - JOUR
AU - Móricz, Ferenc
TI - Tauberian theorems for Cesàro summable double sequences
JO - Studia Mathematica
PY - 1994
VL - 110
IS - 1
SP - 83
EP - 96
AB - $(s_{jk}: j,k = 0,1,...)$ be a double sequence of real numbers which is summable (C,1,1) to a finite limit. We give necessary and sufficient conditions under which $(s_{jk})$ converges in Pringsheim’s sense. These conditions are satisfied if $(s_{jk})$ is slowly decreasing in certain senses defined in this paper. Among other things we deduce the following Tauberian theorem of Landau and Hardy type: If $(s_{jk})$ is summable (C,1,1) to a finite limit and there exist constants $n_1 > 0$ and H such that $jk(s_{jk} - s_{j-1,k} - s_{j-1,k} + s_{j-1,k-1}) ≥ -H$, $j(s_{jk} - s_{j-1, k}) ≥ -H$ and $k(s_{jk} - s_{j,k-1}) ≥ -H$ whenever $j,k > n_1$, then $(s_{jk})$ converges. We always mean convergence in Pringsheim’s sense. Our method is suitable to obtain analogous Tauberian results for double sequences of complex numbers or for those in an ordered linear space over the real numbers.
LA - eng
KW - double sequence; convergence in Pringsheim's sense; summability (C,1,1); (C,1,0) and (C,0,1); one-sided Tauberian condition of Landau and Hardy type; slow decrease; ordered linear space; Cesàro summability; Tauberian theorem; complex-valued sequences
UR - http://eudml.org/doc/216100
ER -

References

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  1. [1] G. H. Hardy, Divergent Series, Univ. Press, Oxford, 1956. Zbl0897.01044
  2. [2] E. Landau, Über die Bedeutung einiger neuerer Grenzwertsätze der Herren Hardy und Axer, Prace Mat.-Fiz. 21 (1910), 97-177. Zbl41.0241.01
  3. [3] I. J. Maddox, A Tauberian theorem for ordered spaces, Analysis 9 (1989), 297-302. Zbl0677.40003
  4. [4] F. Móricz, Necessary and sufficient Tauberian conditions, under which convergence follows from summability (C,1), Bull. London Math. Soc., to appear. Zbl0812.40004
  5. [5] R. Schmidt, Über divergente Folgen und lineare Mittelbindungen, Math. Z. 22 (1925), 89-152. Zbl51.0182.04
  6. [6] A. Zygmund, Trigonometric Series, Univ. Press, Cambridge, 1959. Zbl0085.05601

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